Effective adaptation of hexahedral mesh using local refinement and error estimation

Transcription

1 Key Engineering Materials Vols (2003) pp online at (2003) Trans Tech Publications, Switzerland Online Citation available & since 2003/07/15 Copyright (to be inserted by the publisher ) Effective adaptation of hexahedral mesh using local refinement and error estimation Yoshitaka Wada 1, Mamtimin Geni 2, Masanao Matsumoto 3 and Masanori Kikuchi 4 1 Dept. of Mech. and Systems Design, Tokyo University of Science, Suwa, Toyohira , Chino, Nagano, , Japan 2 Dept. of Mech. Eng., Xinjiang University, Urumqi, Xinjiang Province , PR China 3 School of Mech. Eng., Tokyo University of Science, Yamazaki 2641, Noda, Chiba, , Japan 4 Dept. of Mech. Eng., Tokyo University of Science, Yamazaki 2641, Noda, Chiba, , Japan Keywords: FEM, Adaptive Analysis, Local Mesh Refinement, Hexahedral Mesh Generation Abstract. 3-D finite element method (FEM) is widely used as an effective numerical simulation technique. In the simulation technique, accuracy is one of the most significant issues. In case of FEM, the accuracy is affected by number of freedom and shape of each element. Generally, a fine mesh can provide more accurate result than a coarse one, and needs more time of calculation and more computer resources (memory, CPU time and disk space). Recently tetrahedral automatic mesh generation and adaptive mesh generation become advanced and practical. Hexahedral mesh generation and its adaptation are not enough to use for practical applications, because its mesh generation is very difficult and still intensive labor work by hand. In recent years local mesh refinement for a tetrahedral element is widely used in order to avoid failure of mesh regeneration. Therefore one of the best ways to control quality of a hexahedral mesh is applying local mesh refinement to existing hexahedral mesh. In this study we present a method to generate an appropriate mesh for user's demand using existing hexahedral mesh and hexahedron mesh and hexahedral automatic local refinement technique. Introduction Adaptive technique is one of the most important issues in order to realize effective FEM analysis task. Many works for adaptive mesh generation and error estimation are conducted. But there is a problem that is very complex implementation of such kind of systems. Mesh generation is difficult issue, because there is no guarantee to succeed in a complete mesh generation systems with an arbitrary shape and node distribution. Adaptive mesh generation is highly required on CFD area. Much work has been progressed for adaptive mesh generation for CFD, especially aerospace area. Structured mesh or tetrahedral unstructured mesh are well utilized. Structured grid is easily obtained without fail. On the other hand, unstructured hexahedral mesh refinement is very difficult and challenging problem. Hexahedral mesh is still required for mechanical analysis, especially contact and large deformation problems. Essentially hard geometrical restriction to fill in arbitrary domain exists in hexahedral mesh generation. Hexahedral mesh generation is a time consuming process rather than structured and tetrahedral grids. Since mesh regeneration process requires information on object geometry from CAD system or geometrical curved surfaces, the process demands much time and cost in programming and execution of application. We propose a simple and effective hexahedral mesh adaptation with local mesh refinement. The method is based on modified octree-based hexahedral mesh generation [1] and error estimator [2]. The refinement method which is based on local mesh refinement utilizes original mesh connectivity. All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of the publisher: Trans Tech Publications Ltd, Switzerland, (ID: /01/08,07:43:48)

2 28 Progress in Experimental and Computational Mechanics in Engineering 2 Title of Publication (to be inserted by the publisher) An original method is proposed for fully automatic hexahedral mesh generation with isomorphism technique [3]. The octree-based generation is used for filling the interior of the object with arbitrary density. We find out that the octree-based can control density of elements and it is very useful for adaptive and multigrid mesh generation [4]. We present the effectiveness of adaptive mesh refinement. Presented technique is implemented as a part of GeoFEM [5] software. Refinement of Hexahedral Mesh Figure 1 shows hexahedral refinement processes using modified octree-based method. Figure 1 (a) shows an initial cubic mesh and Figure 1 (b) and (c) represent instruction of which element to be refined and result of first refinement. Figure 1 (d) shows second result of refinement. As shown Figure 1 hexahedral mesh refinement can be applied to any elements. To realize reliable refinement, refinement of patterns is very important. The octree algorithm is a well known to represent geometric shape or features or to optimize and reduce quantities of computational search. The octree structure used here is modified one. One octree structure has 27 sub-octree which is called octant. This 27-tree octant is suitable for hexahedral mesh refinement, because an octant can be easily connected to sub-octants using patterns as shown in Figure 2. P4 is a pattern located in the center of refinement. P3 is a pattern which faces each surface of P4. Six patterns of P3 should be ordinarily generated around one P4. P2 is a pattern which is generated beside P3. Finally P1 is a pattern to be generated beside elements of P2. (a) Initial (b) First Level (c) Second Level (d) Final Level Figure 1. Recursive Mesh Refinement (a) Pattern 1 (b) Pattern 2 (c) Pattern 3 (d) Pattern 4 Figure 2. Refinement Patterns The modified octree-based refinement has three steps as follows. Octree level is assigned to which elements should be refined. Higher octree level means finer mesh to be generated. Then the octree structure is examined to apply mesh refinement patterns without fail. In other words, information for refinement patterns to be rotate and locate properly are generated and examined. Finally, refinement patterns are applied to a mesh.

3 Key Engineering Materials Vols Journal Title and Volume Number (to be inserted by the publisher) 3 Essentially the 3-D process is the same as 2-D. Only difference is number of patterns for each case. In case of 2-D number of refinement patterns is 3. In case of 3-D number of refinement patters is 4. Error Estimation Zienkiewicz and Zhu [2] proposed a simple and efficient residual h-based error estimator in The authors obtained an improved approximation of the discontinuous finite element stress field by a global least squares fit of a piecewise continuous field. These stresses are considered improved, since normally stress equilibrium should exist across inter-element boundaries. We choose the Zienkiewicz-Zhu (ZZ) method, because test case is a crack analysis of uniform material and linear stress analysis. ZZ estimator is enough for linear stress analysis and simple implementation and widely used as standard. An application program of error estimator is independent from analysis code and local refinement application. The estimator can be easily changed to another one, because appropriateness of error estimation method depends on a kind of analysis. Adaptive Analyses for Structural Problems Adaptive analysis using ZZ estimator and local mesh refinement is illustrated as shown in Figure 3. L2 error norm is given by eq. (1). σ is a discontinuous finite element stress function. σˆ is an interpolated continuous stress function. Finally, mean least square of L2 error norm is given by eq. (2). The error norm is chosen as an error index. e 1 T [ ( σ ˆ) σ ( σ ˆ) σ ] 2 = 2 dω σ (1) L 2 e σ L σ = 2 (2) Ω Index of decision of adaptation is given by eq. (3). σ (e) is a mean least square norm given by element e. Elements with η over 10 % are refined in the example. η = 100 σ ( e) σ (3) A model of example is a quarter model of a plate with a hole loaded by uniform stress in z direction. Mesh refinement is applied in the stress concentrated area. Figure 4 shows convergence of the error norm. Error norm is converged at second level adaptation. This index just represents average max error norm over the each element. σ, maximum stress of z σ, of second adaptation level is z max σ of final adaptation level is Because adaptation area becomes narrower and narrower, area z of high error norm becomes narrow and it does not affect whole error norm.

4 30 Progress in Experimental and Computational Mechanics in Engineering 4 Title of Publication (to be inserted by the publisher) (a) Initial Mesh (b) 1st Level Adaptive Mesh (c) 2nd Level Adaptive Mesh (d) Final Level Adaptive Mesh Figure 3. Adaptive Analysis and Mesh Refinement

5 Key Engineering Materials Vols Journal Title and Volume Number (to be inserted by the publisher) 5 Error Norm Number of Nodes Figure 4. Comparison between number of nodes and error norm Future Work Currently mesh refinement process is one way process. In short, mesh is just refined and is not derefined in the current implementation. When physical variables dynamically change in the dynamic analysis, mesh should be refined and derefined in every time step. Octree-based mesh refinement has a hierarchical structure. Refining pattern and transition patterns have to be applied to each hexahedral element. In other words, original hex elements contain refined element patterns. In order to represent hierarchical structure new FE data format should be defined. Practically we should develop a new data format and its application programming interface. The final convergence value of error norm depends on the initial refinement area. Local mesh refinement provides reliable fine mesh but discontinuous mesh size, which is one-third in length. Initial refinement greatly influences convergence of average error norm, because normally area of high error norm exists in the refined area. If area of initial mesh refinement is local, average of error norm converges to higher value than appropriate initial mesh refinement. Therefore it is very important to decide area of initial refinement. When area of local refinement is decided, number of nodes in the next refinement can be calculated. The feature is an advantage for precise controlling mesh. Computation of numerical analysis is restricted by computer resources, which are CPU speed, CPU time, size of memory and so on. As everything of user s demand and computer environment is known, an appropriate mesh, which is optimized for user s request and computer resources, can be generated using the error estimator and the local refinement. Conclusion Reliable adaptation technique is proposed. The adaptation technique is based on a stress error estimator and local hexahedral mesh refinement. The refinement cannot change size of elements continuously, but it can generate refined mesh without failure and number of new generating nodes is easily estimated. It is possible to develop a system for generating appropriate mesh for user s demand and limitation of computer resources by means of proposed adaptive technique. Acknowledgement Local refinement program (HT-R) is a part of products of the GeoFEM project. The authors would like to express their appreciation for the GeoFEM project and Research Organization for Information Science and Technology.

6 32 Progress in Experimental and Computational Mechanics in Engineering 6 Title of Publication (to be inserted by the publisher) Reference [1] R. Schneiders, R. Sxhindler and F.Weiler (1997), Octree-based Generation of Hexahedral Element Meshes, 6th Int. Meshing Round Table, [2] O.C.Zienkiwicz and Z. Zhu (1987), A simple error estimator and adaptive procedure for practical engineering analysis, Int. J. numer. Meth. Eng. 24, pp [3] R. Schneiders (1995), Automatic generation of hexahedral finite element meshes, Proc. 4th Int. Meshing Round Table, pp [4] Y.Wada (200), Effective Adaptation Technique for Hexahedral Mesh, 2nd ACES Workshop Abstract, pp [5] GeoFEM Web site: